Q. Develop a recurrence for H(n), and show why it counts the number of strings correctly.

2. Relevant equations

3. The attempt at a solution

I have tried to form some kind of pattern but I am still not unable to form a recurrence.
we intuitively know that H(n=0) and H(n=1) are both 0 which is still even.
and H(n=2) = 2 since ( 00, 11) takes place.

I was thinking of setting basecase of recurrence upto n = 4.
so list

H(n)
2 n = 2
3 n = 3
5 n = 4
? n > 4

First of all, am I right to list basecases starting from 2?
and what are some crucial steps I need to take in order to find the right recurrence formula

Q. Develop a recurrence for H(n), and show why it counts the number of strings correctly.

2. Relevant equations

3. The attempt at a solution

I have tried to form some kind of pattern but I am still not unable to form a recurrence.
we intuitively know that H(n=0) and H(n=1) are both 0 which is still even.
and H(n=2) = 2 since ( 00, 11) takes place.

I was thinking of setting base case of recurrence up to n = 4.
so list

H(n)
2 n = 2
3 n = 3
5 n = 4
? n > 4

First of all, am I right to list base cases starting from 2?
and what are some crucial steps I need to take in order to find the right recurrence formula

Helps are much appreciated, thank you

First of all, your problem does not appear to be fully described. I had to read though it several times to figure out what you're supposed to accomplish here.

1. You (character) strings are composed entirely of 0's (zeros) and 1's (one's).

2. Each string has an even number of 1's in it.

3. The function, H(n), counts the number of unique n-character strings which can be composed in this manner.

Q. Develop a recurrence for H(n), and show why it counts the number of strings correctly.

2. Relevant equations

3. The attempt at a solution

I have tried to form some kind of pattern but I am still not unable to form a recurrence.
we intuitively know that H(n=0) and H(n=1) are both 0 which is still even.
and H(n=2) = 2 since ( 00, 11) takes place.

I was thinking of setting basecase of recurrence upto n = 4.
so list

H(n)
2 n = 2
3 n = 3
5 n = 4
? n > 4

First of all, am I right to list basecases starting from 2?
and what are some crucial steps I need to take in order to find the right recurrence formula

Helps are much appreacited, thank you

Is there a reason you are not counting 1010, 0101 and 1001? These also have an even number of 1s.

Is there a reason you are not counting 1010, 0101 and 1001? These also have an even number of 1s.

RGV

Sorry I worded poorly..
I should say that substrings of 1's should be even. (given 0 is a delimeter)
eg 10111 is broken down into 1 and 111 (all odd)
1011 is 1 and 11 (odd / even) but all has to be even so this is not right.